Completely Integrable Gradient Flows

Commun. Math. Phys. 147, 57-74 (1992) Communications in

MathematicalPhysics

© Springer-Verlag 1992

Completely Integrable Gradient Flows

Anthony M. Bloch1'*, Roger W. Brocket!2'** and Tudor S. Ratiu3'***1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA2 Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA3 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

Received December 6, 1990, in revised form July 11, 1991

Abstract. In this paper we exhibit the Toda lattice equations in a double bracketform which shows they are gradient flow equations (on their isospectral set) on anadjoint orbit of a compact Lie group. Representations for the flows are given anda convexity result associated with a momentum map is proved. Some generalproperties of the double bracket equations are demonstrated, including a discussionof their invariant subspaces, and their function as a Lie algebraic sorter.

0. Introduction

In this paper we present details of the proofs and applications of the results onthe generalized Toda lattice equations announced in [7].

One of our key results is exhibiting the Toda equations in a double bracketform which shows directly that they are gradient flow equations (on their isospectralset) on an adjoint orbit of a compact Lie group. This system, which is gradientwith respect to the normal metric on the orbit, is quite different from the repre-sentation of the Toda flow as a gradient system on R" in Moser's fundamental paper[27]. In our representation the same set of equations is thus Hamiltonian and agradient flow on the isospectral set.

While the double bracket equations mentioned above are fundamental indemonstrating properties of the Toda flow, they are in fact more general and areof interest in their own right. These equations arose originally in the study byBrockett (see [10 and 11]) of the steepest descent equations corresponding tocertain least squares matching and sorting problems. In [5] it was noticed that a

* Supported in part by NSF Grant DMS-90-02136, NSF PYI Grant DMS-9157556, and a SeedGrant from Ohio State University** Supported in part by AFOSR grant AFOSR-96-0197, by U.S. Army Research Office grantDAAL03-86-K-0171 and by NSF grant CDR-85-00108*** Supported in part by NSF Grant DMS-8922699

58 A. M. Bloch, R. W. Brockett and T. S. Ratiu

particular case of these equations give the Toda (sl(ή)) equations in the Flaschka([15]) form. In fact it can be shown that these equations have many differentinvariant subspaces as we prove here.

Just as it can be shown that the time-1 map of the generalized Toda flows isequivalent to the QR algorithm for diagonalizing symmetric matrices (see [14, 25,30 and 31], for example), these double bracket equations provide a method forsolving the symmetric eigenvalue problem, which coincides with the QR algorithmin the tridiagonal case. Recall also that the Toda equations in Flaschka's form sortthe eigenvalues of a given symmetric matrix into ascending order. Here we showthat the double bracket equations give a general Lie-algebraic sorter, which willsort in any prescribed fashion (corresponding to a choice of Weyl chamber).

We also show here how the equilibria of the double bracket equations are thevertices of a convex polytope which is the image of a momentum mapping associ-ated with the problem (see the work of Schur-Horn [19], Kostant [21], Atiyah[3] and Guillemin and Sternberg [17]). The image of the isospectral Toda orbitis shown to be a complex object lying in the interior of the polytope defined bythe momentum mapping. That a convex polytope may be associated with theToda flows, was observed originally by Deift Nanda and Tomei [14] and Tomei[33] (see also van Moerbeke [34], Fried [16] and Davis [13]). In a related paper[6], by exploiting the Kahler structure of the problem, we show that the isospectralToda orbit may in fact be mapped onto the polytope by a momentum mapping,thus putting the topological observation of Tomei et al. into a symplectic context.

The outline of this paper is as follows. In Sect. 1 we discuss the double bracketequations, indicate how they arise as gradient flows and describe certain of theirinvariant subspaces. In Sect. 2 we show how a special case of these equations givesthe generalized Toda lattice equations (see Kostant [22, 23], Symes [30] andothers). A Lie algebraic explanation is given for the double bracket form of theequations in the Toda case. In Sect. 3 we give explicit representations for the Todaflows in double bracket form for the classical Lie algebras. In Sect. 4 we discussthe convexity and momentum map result and the sorting result mentioned above.

1. The Double Bracket Equations and Gradient Flows

In this section we begin by discussing gradient flows on compact Lie groups. Wederive an explicit equation for a particular gradient flow relative to a left-invariantRiemannian metric. We then show that projecting the flow onto an adjoint orbitof the Lie algebra by taking the natural projection yields a flow which is itselfgradient with respect to the "normal" metric on the orbit. This flow is shown tobe of the "double bracket" form L(t) = [L(ί), Wt\N~\~\ where N is a given constantmatrix. This flow is isospectral and we show that in fact this double bracket flowis the compatibility condition for the "spectral equations" for the flow at the grouplevel. This is related to the classical theory of integrability of Hamiltonian systemsas we shall see in the next section. Finally, we discuss some invariant subspacesof the double bracket equation.

Let Gu be a compact Lie group with Lie algebra &u and let /c(,) denote theKilling form on &u. Define the smooth function F(θ) on Gu by

= ιc(β,AdβΛO (1.1)

Completely Integrable Gradient Flows 59

where Q and N are fixed elements of &u and Ad denotes the adjoint action of Gu

on &u. Then we have (see [10] for the unitary case)

Proposition 1.1. The gradient flow of F(θ) relative to the left-invariant Riemannianmetric on Gu whose value at the identity is minus the Killing form is given by

)-ιβ,ΛΓ], (1.2)

where Θ P denotes the left translation of P by θ.

Proof. Let vθ = θ-R where Re^u and denote by <•,•> the metric on Gu obtainedby left-translation of — κ(v). Then

F(θ exptR)

= ιc([ΛΓ,Adβ-1β],Λ)

= -<0 [N,Adβ-ιβ;

whenceιβ,ΛΓ|. D

Let <gu be the compact real form of a complex semisimple Lie algebra 0. Thenthe equation (1.2) have the normal real form of ̂ intersect the compact real formas an invariant submanifold. The gradient flow can then be formulated as a gradientflow with respect to the normal metric on this intersection. Equation (1.2) firstappeared in this form on S0(n) (see [10]).

Corollary 1.2. The projection of the flow (1.2) onto the adjoint orbit containing β,obtained by setting L(t) = Adβ(ί)- 1 β is given by

L(i) = [L(i),[L(i),JV]]. (1.3)

Indeed, by the Chain Rule and Proposition 1.1 we get

It is interesting to notice that Corollary 1.2 can be interpreted as a compatibilitycondition between an "eigenvalue equation" and the time evolution of the"eigenfunction." Indeed L(t) = Adθ(ί)-ι β says that L(t) and β have the samespectrum, if they are matrices. In other words, this relation can be thought of asthe totality of eigenvalue equations for L, with the columns of θ as eigenfunctions.The time evolution for θ is given by (1.2). The usual cross-differentiation argumentrepeats the proof of Corollary 1.2.

Corollary 1.3. The Eq. (1.3) is the compatibility condition for the system L(t) =Adθ(ί)- 1 β and θ(t) = θ(t) [L(ί), ΛΓ|.

Note that the flow (1.3) is the image of (1.2) under the quotient map Gu -> GJH,H the stabilizer of L(0) = L0, where GJH is identified with the adjoint orbit θ


through L0. This "projected" flow turns out to be itself a gradient flow on theadjoint orbit 0 endowed with the "standard" or "normal" metric (see Atiyah [3]).Explicitly this metric is given as follows.

Decompose orthogonally, relative to - κ( ,) = <,>, <$u = <9^ © ̂ uL, where ^ttL

is the centralizer of L and ̂ = Im ad L. For Xe&u denote by XLe^ the orthogonalprojection of X on ̂ . Then set the inner product of the tangent vectors [L, X~\and [L, Y~\ to be equal to <AΓL, YLy. Denote this metric by <, >Λr. Then we have

Proposition 1.4. The flow (1.3) is the gradient vector field of H(L) = κ(L, N\ K theKilling form, on the adjoint orbit (9 of <$u containing the initial condition L(0) = L0,with respect to the normal metric <( , )N on (9.

Proof. We have, by the definition of the gradient,

dH [L, δL] = <grad H, [L, δL] >N, (1.4)

where denotes the natural pairing between 1 -forms and tangent vectors and[L,(5L] is a tangent vector at L. Set grad/f = [L,̂ ]. Then (1.4) becomes

y = < [L, xι [L, δL] yN

or

Thus

andgrad// = [L,[L,ΛΓ|]

as required. Π

The "double bracket" flow (1.3) is in the "Lax pair" form L = [£,L] and isthus isospectral. This of course plays a key role in our Toda lattice analysis.

Now for the Toda lattice analysis we need to consider the invariant subspacesof the equation L = [L, [L, JV]]. We prove here some general results regarding theinvariant subspaces of this equation. For this analysis we may take L and N tobe the elements of any Lie algebra ^ over the real or complex field. -The differentialequation L = [L, [L, N]] then evolves in .̂

Proposition 1.5. Let reR and let & be a Lie algebra. Let Ne& and suppose (ad Λ/)2

has an eigenvalue r2 with corresponding eigenspace V in <$. Let $N be the centralizer ofN. Then ^N@V is an invariant subspace for the nonlinear operator on & given by

Proof. Set V=Vr®V_r, where Vr and K_ r are the +r and -r eigenspaces,respectively, of the operator ad N. We can write a typical element of $N φ V asv++v_+h with [/z,ΛΓ|=0, [Λf, ι> + ] = π; + and [JV,ι;_] = — rt;_. By the Jacobiidentity, we have [N, [>+,*;_]] = 0, and [N, [Λ,tf+]] = r[h,v+~\ so that [h,v+~\evr.Similarly, [fe,t;_]et;_Γ. Together these identities imply

[υ+ + v. + ft, [ι?+ + u_ + ft, JV]]6^φ K D

We also have the following.

Proposition 1.6. Suppose e is an involutive Lie algebra automorphism of^. Let ̂ even

and ôdd be the + 1 and — 1 eigenspaces of e. Then ifadN maps ôdd into êven, itfollows that (&N® JΌπôdd is an invariant subspace for the operator [ ,[ ,N]].


This follows from Proposition 1.5 and the obvious relation [^odd, ̂ even] ϋ ̂ odd.The Toda flows will be seen to evolve in a subspace of the form defined in

Proposition 1.6 with r2 = 1.

2. The Double Bracket Equations and the Toda Lattice

In this section we show that the generalized Toda lattice equations may be writtenin the double bracket form (1.3) and hence are a gradient flow on their isospectralset.

Let &n be the normal real form of .̂Then the Flaschka form of the generalized Toda lattice equations associated to

&n are given by A = [#, A~\ where

i iA= Σ*A+ Σ */*., + *-«> (2 !)

j = l j = l

I

B= ΣM^-e-.Λ ^ >° (2 2)j=ι

This form of the Toda lattice equations is the one adopted in Moser [27] for&n = sl(n,JR.\ and is equivalent to, but different from, the conventions used inKostant [22] and Symes [30, 31]. This is a Hamiltonian system on the coadjoint

iorbit of the lower Borel subalgebra of &n through £ (eΛj + e_Ά.) called the Jacobi

j=ίorbit, with Hamiltonian function ^κ(A,A). All the elements of this orbit which areof the form (2.1) we call Jacobi elements. This system is integrable; the integralsare the basis of the ring of invariants of &„ restricted to the orbit. Level sets of theintegrals of motion are called isospectral sets.

In what follows, let $u be a compact Lie algebra, jtf a maximal abeliansubalgebra and ^ the complexification of ^u, and choose tf = jtf ® i<$/ as theCartan subalgebra of ̂ . We let Φ denote the system of roots of ̂ defined by J f,let Δ denote the simple roots and choose {hp eΛ\j = 1,..., /, αeΦ}, a Chevalley basisof ,̂ with hjeA. α 1 ? . . . , α j denote this simple roots.

Let G and Gu denote the Lie groups with Lie algebra ^ and $u respectively.

Theorem 2.1. If N is i times the sum of the simple coweights of^, then, for

ί =Σ*A+Σ'βA, + *-.,)> (2-3)J = l 7 = 1

Eq. (1.3) gives the generalized Toda lattice equations on the level set of all the integralsof the Toda flow. Explicitly, N is given by

N = Σi*jhp (2.4)j

where (x1,x2? ?:)cί) ίs the unique solution of the system

)=-l, i =!,.. .,/. (2.5)


Proof. From (2.3) and (2.4),

[L,ΛΠ=Σ Σ*A α«^-e-αi)ί=U=ι

But for the generalized Toda flow, L = [£, L] where B is given by (2.2)It is known that there is a unique element N in the Cartan subalgebra (see

[20], p. 1014) such that [N, eΛ~] = height (α)βα. This element is found in the followingΛ

way. Let αf = - — , where (,) denotes inner product, be a dual root and let λh

(a/,af) f

where (!i? α,-) = δij9 be the corresponding dual weight. Set h# = £ lf. Theni = l

(/^? £aj = 1 and we set N = - ίh$. Note that N is the Cartan part of a principal s/(2).Explicitly, [L, AT] = —5 if and only if the coefficients xt in (2.4) are the solution

of the systemi

Σ xj*i(hj)=-lj=ι

It is a standard result that α^/i,-) is the Cartan matrix of .̂ Π

We list now the coefficients xt (i.e. the solutions of the equation Cx = — 1, Cthe Cartan matrix for all the semi-simple Lie algebras (see [8]):

Al

C,

G2 X! = — 3, x2 = — 5

F4 χ1 = -ll, x2=-21, x3=-15,

£6 X! = -8, x 2 = - l l , x3=-15,

x4=-21, x5=-15, .x6 =-8

£7 X j = — 17, x 2

= ~ ¥ ' X3 = ~ 33,

x4=-48, x 5 = : -¥' x6=-26,

£8 χ1 =-46, x2=-68, x3=-91,

x4= -135, x 5 = -110, x 6= -84,

x 7=-57, xβ=-29.

Table 1


Now we can show quite intrinsically that Eq. (1.3) with L and N given by(2.3) and (2.4) respectively are the Toda lattice equations; that is, the TodaHamiltonian on a coadjoint orbit of the lower Borel subalgebra of <&n gives rise tothe Eq. (1.3)

We first derive an explicit formula for the Kostant-Kirillov-Souriau (orbit)form on the coadjoint orbit of the lower Borel subalgebra. Before we do this weneed some notation and definitions:

Let Φ, Φ+, and Φ~ denote the system of roots, the positive and negative rootsof ^ respectively. Let

U=l .

be the normal real form of .̂Now consider the decomposition

where

9Λ, (2.7)αeΦ~

^α the 1R span of the α root space, and let

^ = { Σ cα(eα-e_α)|cαe!RJ. (2.8)UeΦ + J

Then the complements of 3? and Jf under — κ( , ) are

^?1 - U ̂ (2-9)<zeΦ~

and

jf-L = iΛ/ej Σ dβ(ββ-e_β)|dβ6RJ. (2.10)UeΦ+ J

Also

g^g* = 3>-L@jfλ (2.11)

andJSf^jf1, Jf*^^1. (2.12)

Let π^, π^, π^i and π^± denote the associated projections.Now consider the tridiagonal elements of Jf and Jf1, i.e. if Δ denotes the set

of simple roots of ,̂ set

(2.13)

(2.14)UeJ J

J

Let M = — L/VezW, i.e. M = ^ X y / i y , where x l 5 . . . , xt is the solution of the linear

system (2.5). j=1


Then adj^ jΓ^JΓ1©^ is the jsomorphism sending an element of Jf* tominus its symmetrization and ad^Jf"1-* Jf* sends an element AeJfλ to minusits skew-symmetrization (i.e. to —B in (2.22)) and has kernel equal to istf. ThusadM : Jf -1 θ isύ -> tf is also an isomorphism. The map ad^ : JΓ λ -* Jf ^ Q istf is theprojection.

For a fixed ΛeJΓ1 with all αα > 0, αezl (see (2.1)), the operator adA:is/-+ JΓ isan isomorphism. Indeed, proceeding as in the proof of Theorem 2.1 and denoting

we have [A, Z] = P if and only if

(2.16)

(2.17)

Denote by ad^^JΓ-^/j/ the inverse of this isomorphism.The operator ad[A M]\i£# -+tfL Qistf is also an isomorphism, for if Z is as in

β = Σ (2.18)

then [[/4,M],Z] = β if and only if (2.17) holds with pΛ. replaced by qΛι. Let adt~|M]:istf denote the inverse of the isomorphism.

Theorem 2.3. The orbit symplectic form on a co-adjoint orbit of the lower Borelsubalgebra of^n is given by

ω = \(άA Λκ(ad; ί + aά~M\A] adM)adM dA) (2.19)

where Λ K is the wedge product associated with the bilinear form K.

Proof. For X,Ye^, the orbit symplectic form on a Jacobi orbit is given by

Consider then

\(άA Λ l, 7])

Since

and

, XI (ad

, 7] = - π j

(adM)2 πjr^A, F] =

adM) ad , 7])

the above equals:

i/c([/l, XI -(ad-^π

-|κ(OL, 7], -(

, ad; x π^

, 7] - diag [A,

+


= $K(X,nirlA, TΓ\ -aάAaά-M\A^d2

MπÂ, TΓ\)

- i κ( y, π jr [X, Λ ] - adA aά~M\A} ad2

M nÂ.

where ad{M

1

A]adlfπχ ί[A,Y] = Z, and ad^âd^ nχί\_A,X~\ = Z'εistf, and wealso have Z = — diag Y,Z' = — diag A", projections into is/ of 7 and X respectively.Therefore we get

i κ(X, [A, diag ΓJ) - i κ(y, [A, diag A"])

as required. Π

Then we have the

Theorem 2.4. For the orbit symplectic form given in Theorem 2.3 the equations ofmotion on the Jacobi coadjoint orbit of the lower Borel subalgebra of^n correspondingto the Hamiltonian \κ(A,A) are A = [A,[A, — M]].

Proof. ω([/4,[^,

= \κ(lA,lA, -MJKad;

ad;1 +ad[7/%1adJW)adM([A)[/l, -M]])

where Z,Z',Z"eij^ are uniquely determined by the conditions

IA,Z] = IM,[A,IA, -M]]], [[M,/1],Z'] = ad^πjr,U,X],

and [ [M, /I], Z"] = ad^ [/!,[/!, -M]]. Since [/1,M] = B given by (2.2) andad2

M erases the Cartan part of elements in Jf1, the last equation is equivalent toΓ_#,Z"] = ad^[Λ,.β] which in turn determines Z" = -diag A Thus we get

i κ(π jf i IA, X], A - diag A) + \κ( [_X, A~\, Z)

. D


We remark that the Toda lattice is gradient on its restriction to the isospectralset, since it is the restriction of a gradient vector field on a Gu orbit. There appearsto be no appropriate metric off the isospectral set. Similary, Moser's form of thegradient flow (see [27]) also occurs on the isospectral set and is given by

dr±_ dV

at drk

wheren

Σ Vk

2L~t K K

2 Σ ' 2

fc=l

Here the λt are constant and satisfy λ^ < λ2 < - - < λn9 and the rk are strictly positiven

and satisfy £ rl = l For the precise relationship between Moser' flow and the

double bracket flow see [6].We remark also that for s/(/+ 1) our function κ(L9N) is (modulo a trivial

normalization) the Morse function discussed in [13,16 and 33] and used to analyzethe topology of isospectral sets of Jacobi (symmetric, tridiagonal) matrices. Wehave shown that this Morse function gives the Toda flow as its gradient flow withrespect to a suitable metric, and, moreover, we have given a naturalgeneralization of this function to arbitrary semisimple Lie algebras.

We note also that in view of the above analysis, it is natural to write thePoisson bracket on Jacobi elements as

{/(L), g(L)} = -κ(L, [V/ - \_N, V/], Vg - [N9 Vg~]\

where V/ is the gradient of / in <Sn defined by df(L)-δL = -κ(V/,<5L), δLe9n.This is equivalent to the bracket given in Symes [29].The overall picture we have developed is as follows. The Toda flow is

Hamiltonian in the Jacobi elements embedded in &u via (2.3). The level sets of theintegrals of motion are Lagrangian submanifolds of the set of Jacobi elements.Also the level sets lie on orbits of &u. These level sets are invariant submanifoldsfor the gradient flow of the function κ(L, N) with respect to the normal metric.On the level sets the Hamiltonian and gradient flows are identical for N as inTheorem 2.1.

Remark. We note also that one can compute quite explicitly the gradient flowwith respect to the normal metric of an arbitrary function φ(A) on an orbit Θpassing through A0e&un = ̂ un^π. Let V denote the gradient relative to the non-degenerate bilinear form <, > = — κ(,) on &„. Then the flow is given by

A = [A, [A, - V(/>]]. (2.20)

The proof is essentially the same as that of Proposition 1.4.Further, if /:^Π^R is an invariant function, i.e. [VI(A), A] = 0, then the

gradient flow of φ(A) = —κ(VI(A),M), where M — —IN, is

(2.21)

This follows from (2.20) together with the observation that taking the derivative


on <$n at A in the direction of Meis/ of the relation [VI(A\ A] = 0 givesld(yi)(A) M9 A] + [V/04), M] = 0.

3. Representations for the Toda Flows

In this section we consider representations for the Toda flow in the double bracketform for the classical Lie algebras. The key new ingredient here is of course thedetermination of the matrix N. For convenient bases for the classical Lie algebras,see, for example, Sattinger and Weaver [28], or Humphries [24].

At. We take here h± =diag(l, -1,0,0,...), A2 = diag(0,l, -1,0,...). If Eu is thematrix with 1 in the i/ th entry and zeros elsewhere, the positive simple roots are

From N = Σiχjhj> Table 1, and (2.3), we find;j

= idiag[ ->1-2

b2-bl

at

-bt

(3.1)

(3.2)

Ct. Here we take H1 =diag(l, -1, -1,1,0,...,0), h2 = (0,0,1, -1, -1,1,...),...,h, = (0,0,..., 0,0,1, -1). Now, define the functionals oit on the Cartan subalgebraby <xί(H) = λί, where H = diag(λ1α0,...,λlα0), α0 as in (3.4) below. (Note that α;here is not the same as that used in the definition of Chevalley basis.)

Calculating as before, we find

= ίdiag(i(l-2/), -i(l-2/),i(3-2/), -I(3-2/),..

We now calculate L explicitly. Following [28], define

f,f,

0 1

0 0

0 0

1 0αn =

1 0

0 -1

(3.3)

(3.4)

The simple positive roots are oc1 — α2,..., ot,_ j — α,, 2α, with corresponding rootvectors

α ;-α ;:

2α,:(α+)Hand


with the double subscript indicating the position of the ( i j ) 2 x 2 block. 1 is the2 x 2 identity matrix.

Then for C3 we haveb, 0 a, 0 0 0

0 -b1 0 «! 0 0

-0! 0 b2-b1 0 α2 0L = i

0 -Λ! 0 b1-b2 0 α2

0 0 -a2 0 b2-b3 a3

0 0 0 -a2 a3 b2-b3

Dt. Here the ht are as for Cl

From Table 1

N = ίdiag(-/ +1,/-1,-/ + 2,2-/, -.,-1,1,0).

(3.5)

(3.6)

We again calculate L.Defining the α, as for Q, the simple positive roots are o^ - α2, α2 - α3, α^_ x - αί?

and a t _! + α; with corresponding root vectors

Then for Z)3 we have

L = i

&! 0 αt 0

0 -&! 0 -Λ!

αι 0 b

2 — b

1 0

0 — αλ 0 —b

ί-\-b

2

0 0 α2 α

3

0 0 -α3 -α

2

0

0

0

0

-a2

0

0 b2-b3

(3.7)

βj. Here we take /ιx =diag(0,1, - 1,0,...), /ι2 = diag(0,0,0,1,-1,...), etc. Then

JV = ίdiag(0,-/,/,-/+!,I-/,...,-1,1). (3.8)

We now calculate L.Defining α, here by ai(H) = λi, where // = diag(0,A1α0,...,/ί/α0), the simple

positive roots are

with root vectors:

Completely Integrable Gradient Flows

Thus for B

L =

• o0

0

0

0

-α3

a3

0

fri

0

«1

0

0

0

0

0

0

— ύ

0

0

a2

0

0

0

— α,

0

0

69

0

0

0 (3.9)

4. The Double Bracket Equation, Momentum Maps, and Toda Flows

In this section we discuss how the theory of convexity of the image of the momentummaps, as developed by Schur-Horn [19], Kostant [21], Atiyah [3] and Guilleminand Sternberg [17], may be used to deduce properties of the double bracketequation and hence the Toda lattice flows. We also discuss Lie-algebraic sorting.

As observed by Brockett in [10], one can in fact view the equation L(t) =[L(t\ [L(£),7V]] in the Hermitian case as solving a linear programming problemon the convex set defined by the Schur-Horn theorem. This also makes contactwith the problem of minimizing the Total Least Squares function (see [4 and 11]).

Theorem 4.1. (i) Let Θ be a (co)adjoint orbit of Gu in &u and consider the gradientvector field L(t) = [L(ί), [L(ί), ΛΓ]], L(0)e0, with flow FJor N a fixed regular elementin <£„. The set of equilibria of this vector field coincides with G)r\stf, where stf is theCartan subalgebra of <$u containing N. This set ®r\s0 consists of a single Weylgroup orbit. By Konstant's convexity theorem ([27]), the convex hull of theseequilibria is a compact polytope 0* which is the image of (9 under the momentummapπ\Θ-+£# defined by the adjoint T -action on (9, where T is the maximal torusin Ήu obtained by exponentiating j/ and the momentum map π is the orthogonalprojection of (9 onto <$#. n(Ft(&)) lies entirely in 0*. The number of equilibria of Ft

equals the order of the Weyl group for L(0) regular.(ii) IfN and L are chosen as in Theorem 1 (ii), L(t) = [L(f), [L(ί)>W]] becomes the

Toda lattice equations defined on the level set of all constants of the motion and theprojection of the Toda flow on jtf lies in the interior of the polytope given in (i).

Proof. T acts on 0 by the (co)adjoint action, defining the momentummapπ:0->j/ = IR' which is the orthogonal projection of Θ onto stf (relative tothe metric — κ( , ) on ^u). The image of π is a convex compact polytope which isthe convex hull of the critical values of π. These critical values coincide with theWeyl group orbit 0nj/ (see [21]).

To prove the theorem we will show that the equilibria of the vector fieldL(t) = [L(t),[L(f),JV]] on <3U necessarily lie in j f . (See [10] for the unitary case.)Note firstly that the solutions of the equation L(ί) = [L(ί), [L(ί), N]] exists for alltime since the flow is the projection of a flow from a compact group. We will now


show that lim L(ί) must be in stf. Observe that

(κ(L, N)) = κ(N, [L, [L, TV]]) = - κ([L, ΛΓ], [L,at

0.

Hence κ(L, N) is monotonic increasing along the flow and is bounded above sinceL(ί) lies in the coadjoint orbit of a compact group. Thus κ(L, N) has a limit andits derivative goes to zero. But we can see that its derivative vanishes only if Land N commute and, since N is regular, L must lie in stf. Π

We remark that the Toda lattice equations on the level set of all constants of themotion have no equilibria, but if one considers these equations defined on theclosure of the Toda phase space, then the equilibria are given by the Weyl grouporbit as stated in Theorem 4.1.

We remark that in [14 and 33], it was shown that an isospectral manifold ofJacobi (symmetric, tridiagonal) matrices with the off diagonal elements taken tobe nonzero - i.e. a Toda orbit - is homeomorphic to a convex polytope. In a relatedpaper [6] we show, by studying the Kahler geometry of the Toda flows, that thispolytope is in fact the image of a momentum map.

In this paper we restrict ourselves to observing that while the image of themomentum map discussed in Theorem 4.1(i) is indeed convex, when one restrictsthe domain to the Toda orbit, one obtains an image that lies in the interior of aconvex polytope, with equilibria at its vertices, but the image is far from convex.This is illustrated for A2 below, where the projection onto the Cartan subalgebraof the integral curves of the system with conserved eigenvalues λ1 = 1, λ2 = 2, andλ3 = 3, are shown. (This image has also been constructed by H. Flaschka andM. Zou.)

It is well known (see Symes [31] Deift, Nanda and Tomei [14]) that for theToda lattice equations on s/(n) the only stable equilibrium is that where theeigenvalues of L are sorted into increasing order along the diagonal of L. We canprove the Lie algebraic generalization of this result using our double bracket

1

1 2 3

Fig. 1. The projection of the Toda trajectories under the image of the momentum map for A2


formulation, but also a more general result on sorting which applies to the(non-Toda) double bracket flow on a compact Lie algebra. This result statesessentially that the only stable equilibrium of the double bracket flow is thatwhere L is diagonal and the diagonal entries are the eigenvalues of L sortedcommensurately with N. More precisely, our result (which is a Lie algebraicgeneralization of the Hermitian result in [10]) is

Theorem 4.2. Consider, as in Theorem 4.1, the (cό)adjoint orbit Θ of Gu in &u andthe vector field L(t) = \_L(t), \_L(t\ N] ] on Θ. Assume N is regular. The only stableequilibrium, L0, of this vector field, lying necessarily in 0n«s/, where stf is the Cartansubalgebra of &u containing N, is the (unique) equilibrium which lies in the sameWeyl chamber as —N.

Proof. We prove the result by examining the linearization of the vector field) = [L(ί),[L(ί),ΛΓ]]αίL0.Taking the differential of the vector field at L = L0 gives

δ([L, [L, ΛΓ|]) = IδL, [L0, AH] + [L0, [<5L, ΛΓ|]

= [L0,[<5L,ΛΓ|], (4.1)

since L0 and N commute, L0 being an equilibrium point.Hence the linearized vector field is given by

(4.2)

As in Sect. 2, let {hj9eΛ\j=l9...9l9 αe4} be a Chevalley basis of 0, thecomplexification of the Lie algebra &u. stf is a maximal abelian subalgebra of <$u,J^ = <s/@i<$/ is the Cartan subalgebra of ̂ and h^^ for j= I , . . . ,/, α^...,^denote the simple roots. Then a basis for &u is {ihp xa:=e(X — e_a,yα:= i(ea + e_a)[/ = !,...,/, a a positive root}. Let (,) denote the inner productinduced by the Killing form on roots and let <α,β> = 2(<x, β)/(β, β)eZ. We have

[ihj, xβ] = <α, o0);α and [ihj9 yj = - <α, α,.>xα. (4.3)

Therefore, if

N = Σ *A » ^o = Σ >AJ = l f c = l

/

p=l α > 0

for x7,j;fc,^,cα,dα6]R, we get

α>0,j

= — Σ (x7 cα<α,a/>j;a — XjdΛ(θL,a^>xa), (4.4)a > O J

and hence

[L0,[5L,ΛΓ]]= -Γ Σ W\, Σ Σ (*;C.<«,«J>Λ-xJd.<α,«J>xβ)l|_fc=l α > 0 j = l J

X"1 / \ Γ Ί ~l X^1

α > 0 α > 0


= Σα>0

ι

Σ / v» \ / \-« \ ,( «. Σ */«j ) ( α> Σ Λ«* )(CΛ + ̂ J. (4.5)

α > 0 \ j = l / \ *=1 /

Therefore, the linearized system (4.2) decouples into

i, = 0, P = l , . . . , /

/ ± \/ ± \c« = cx(<x, Σ x j Ά j W α , X ΛoO, α>0\ j=ι / \ t=ι / (4.6)

dΛ = dΛ(ct, Ϋ x α )( α, Ϋ 3;tαt }, α>0.

Now, since the Weyl group acts simply transitively on the Weyl chambers, we

may assume that N lies in the positive Weyl chamber, i.e., ( α, Σ xjαj }>0\ 7=1 /

Therefore, the origin will be a stable equilibrium for the decoupled linear system

(4.6) iff ( α, Σ yΛ } = 0, i.e. iff L0 lies in the reflection through the origin of the\ fc=ι /

positive Weyl chamber, that is, the Weyl chamber of — N. Π

Remarks. 1) If N is not regular, the conclusion of the theorem does not hold any

/ l \more. Indeed, if for some α > 0, ( α, Σ xj^j = ®> i e > N lies on a wall, then in

(4.6) the sign of ( α, Σ >Ά ) cannot be controlled.\ k=l I

2) If 3?u = su(l + 1), the off-diagonal part of (4.6) reduces to

Sty = (λπ(j) - λπ(i})(Πj - njδlij,

where W = diag(w1,...,ιn/ + 1), L0 = diag(ΐλπ ( 1 ),...,ίλπ ( / + 1)) are traceless diagonalmatrices with purely imaginary entries, and π is a permutation of (1, 2, . . . , / + 1}.In other words, in the limit the flow of L(f) = [L(ί), \_L(t\ N~\~] sorts the eigenvaluesof L such that they appear on the imaginary axis in opposite order to the eigenvaluesof N. This result in the Hermitian context was obtained by Brockett [10]; therethe real eigenvalues of N and the limit of L appear in the same order on the realaxis.

Requiring all the coefficients in (4.6) to be positive characterizes the uniqueequilibrium which is a source. We get:

Corollary 4.3. In the hypotheses of Theorem 4.2, the unique equilibrium ofL(t) = [L(ί), [L(i),N]] wfticΛ is a source lies in the same Weyl chamber as N.

Both Theorem 4.2 and Corollary 4.3 generalize to the following.

Theorem 4.4. In the hypotheses and notations of Theorem 4.2, let Ls be the uniqueequilibrium which is a source and let σ(Ls) be another equilibrium, where σ is anelement in the Weyl group. Then the dimension of the stable manifold ofσ(Ls) equalstwice the length of σ.


Proof. From the system (4.6), we see that the dimension of the stable subspace of

the linearized system at the equilibrium σ(Ls) = Σ ykihk, equals twice the numberk=l

of α > 0 such that ί α, Σ ykctk 1 < 0. However, ί α, Σ 3Ίkα* ) = ί σα, σί Σ yA ) )/ i \ \ k=ι / \ k=ι / \ \k = ι //

and σί Σ yΛ } *s in the positive Weyl chamber by construction and choice of\ k = 1 /

ordering. Since the only roots which have positive inner product with an elementin the positive Weyl chamber are the positive roots, we need to determine allpositive roots α for which σα < 0. This number, however, is the length of σ (seeHumphreys [24] p. 52). Π

We remark that Theorem 4.4 can also be proved by examining the gradientflow κ(L, N) with respect to the Kahler metric on 0. This is done in Atiyah [3],where the dimension of the stable manifold is shown to coincide with the dimensionof Bruhat cell association with the equilibrium. This result is equivalent to ours,since the dimensions of the stable and unstable manifolds at the equilibria of agradient vector field are independent of the metric used. This may be seen asfollows. The linearization of the vector field V/ at a given equilibrium is g~lHwhere H is the Hessian of / at the equilibrium and g is the local matrixrepresentation of the metric. The eigenvalues of g~1H are equal to those ofg~1/2Hg~1/2 = (g~1/2)τHg~^2, which is a quadratic form having the same rankand signature as H.

We can thus see that the gradient flow (1.3) has a number of remarkableasymptotic properties. When restricted to Jacobi elements, we have seen theseequations are actually Hamiltonian. Although there are other invariant subspacesfor these equations, as we have shown, we know of no other cases where they areHamiltonian. We note however that a key point which allowed us to show thenonperiodic Toda lattice equations are gradient on their isospectral set is thenoncompactness of the level sets. We suspect that other integrable systems withnoncompact level sets may exhibit gradient-like behavior or indeed be gradientflow on their level sets. We are currently investigating this idea.

Acknowledgements. We would like to thank H. Flaschka for his valuable comments during thecourse of this work and J. E. Marsden for several useful conversations. We would also like tothank J. Lagarias for some illuminating discussions on gradient flows and integrable systems,and the referee, whose suggestions greately improved the exposition.

References

1. Abraham, R. A., Marsden, J. E.: Foundations of mechanics. Benjamin Cummings. 19782. Adler, M: On a trace functional for pseudo-differential operators and the symplectic structure

of Korteweg-de Vries type equations. Invent. Math. 50, 219-248 (1979)3. Atiyah, M. F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 16, 1-15

(1982)4. Bloch, A. M.: Estimation, principal components and Hamiltonian systems. Syst. Control

Letters 6, 103-108 (1985)5. Bloch, A. M.: Steepest descent, linear programming and Hamiltonian flows. Contemp. Math.

AMS 114, 77-88 (1990)6. Bloch, A. M., Flaschka, H., Ratiu, T.: A convexity theorem for isospectral sets of Jacobi

matrices in a compact Lie algebra. Duke Math. J. 61 (1), 41-66 (1990)


7. Bloch, A. M., Brockett, R. W., Ratiu, T.: A new formulation of the generalized Toda latticeequations and their fixed point analysis via the moment map, Bull. AMS 23 (2), 447-456 (1990)

8. Bremner, M. R., Moody, R. V., Patera, J.: Tables of dominant weight multiplicities for repre-sentations of simple Lie algebras. Marcel Dekker

9. Brockett, R. W.: Least squares matching problems. Linear algebra and its applications,122/123/124, 761-777 (1989)

10. Brockett, R. W.: Dynamical systems that sort lists and solve linear programming problems.In: Proc. 27th IEEE Conf. on decision and control, IEEE, 1988, pp. 799-803, subsequentlypublished in Linear Algebra and its Applications 146, 79-91 (1991)

11. Byrnes, C. I., Willems, J. C.: Least squares estimation, linear programming and momentummaps. IMA J

12. Chu, M. T.: The generalized Toda lattice, the QR-algorithm and the centre manifold theory.SIAM J. Alg. Disc. Math. 5, 187-201 (1984)

13. Davis, M. S.: Some aspherical manifolds. Duke Math. J. 5, 105-139 (1987)14. Deift, P., Nanda, T., Tomei, C.: Differential equations for the symmetric eigenvalue problem.

SIAM J. Numerical Anal. 20, 1-22 (1983)15. Flaschka, H.: The Toda lattice. Phys. Rev. B 9, 1924-1925 (1976)16. Fried, D.: The cohomology of an isospectral flow. Proc. AMS 98, 363-368 (1986)17. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67,

491-513 (1982)18. Guillemin, V., Sternberg, S.: On the method of Symes for integrating systems of the Toda

type. Lett. Math. Phys. 7, 113-115 (1983)19. Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math.

76, 620-630 (1956)20. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex

simple Lie group. Am. J. Math. 81, 973-1032 (1959)21. Kostant, B.: On Convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. EC.

Norm. Sup. 6, 413-455 (1973)22. Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math.

34, 195-338 (1979)23. Kostant, B.: Quantization and representation theory. In: Representation theory of Lie groups,

London mathematical society. Lecture note series 34, pp. 91-150. Cambridge UniversityPress 1979

24. Humphries, J. E.: Introduction to Lie algebras and representation theory. Berlin, Heidelberg,New York: Springer 1972

25. Lagarias, J.: Monotonicity properties of the generalized Toda flow and QR flow. SIAM J.Matrix Anal. App. (to appear)

26. MacDonald, I. G.: Algebraic structure of Lie groups, in representation theory of Liegroups, London mathematical society. Lecture note series 34, pp 91-150. CambridgeUniversity Press 1979

27. Moser, J.: Finitely many mass points on the line under the influence of an exponentialpotential. Batelles Recontres. Springer notes in physics, pp. 417-497. Berlin, Heidelberg,New York: Springer 1974

28. Sattinger, D. H., Weaver, D. L.: Lie groups and Lie algebras with applications to physics,geometry and mechanics. Berlin, Heidelberg, New York: Springer 1986

29. Symes, W. W.: Hamiltonian group actions and integrable systems. Physica D 1, 339-376(1980)

30. Symes, W. W.: Systems of Toda type, inverse spectral problems and representation theory.Invent. Math. 59, 13-51 (1982)

31. Symes, W. W.: The QR algorithm and scattering for the nonperiodic Toda lattice. PhysicaD 4, 275-280 (1982)

32. Toda, M.: Studies of a non-linear lattice. Phys. Rep. Phys. Lett. 8, 1-125 (1975)33. Tomei, C.: The topology of isospectral manifolds of tridiagonal matrices. Duke Math. J. 51,

981-996 (1984)34. vanMoerbeke, P.: The spectrum of Jacobi matrices. Invent. Math. 37, 45-81 (1976)

Communicated by N. Yu. Reshetikhin

Completely Integrable Gradient Flows

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